11 Soft Active Materials Zhigang Suo Harvard University II. Gels

2 gel = network + solvent Solid-like: long polymers crosslinked by strong bonds. Retain shape Liquid like: polymers and solvent aggregate by weak bonds. Enable transport solvent network gel Imbibe solvent, and swell Exude solvent, and collapse Ono, Sugimoto, Shinkai, Sada, Nature Materials 6, 429 (2007)

3 Gels in daily life Tissues, natural or engineered Contact lenses Wichterie, Lim, Nature 185, 117 (1960) Drug delivery Ulijn et al. Materials Today 10, 40 (2007) food plants

4 Gels in engineering Gate in microfluidics Beebe, Moore, Bauer, Yu, Liu, Devadoss, Jo, Nature 404, 588 (2000) packer in oil well Shell, 2003 Responsive to physiological variables: pH concentration of ions temperature light

5 Chemical potential of a molecular species in a system vapor A molecular species: water, or a type of alcohol, or a type of protein A system: the wine, or the cheese, or the vapor, or the composite Number of water molecules in a system, M Helmholtz free energy of a system, F(M,…) Chemical potential of water in a system, In equilibrium, the chemical potential of water in the wine equals the chemical potential of water in the cheese, and equals the chemical potential of water in the vapor. Measure the chemical potential of water in one system by equilibrating this system with another system in which the chemical potential of water is known. wine cheese

6 Chemical potential of a species in a one-species system vapor State of reference liquid & vapor in equilibrium liquid vapor liquid onlyvapor only Pressure of saturated vaporvolume per solvent molecule piston weights Relative humidity,

7 The ascent of sap Dixon and Joly, Phil. Trans. R. Soc. Lond. B 186, 563 (1895) The air is dry. The soil is wet. The gradient of humidity pumps water up. Henry Horatio Dixon

Liquid water in tension water  water-permeable solid Chemical potential of water in the air Chemical potential of water in the xylem When the liquid water in the void equilibrate with the water vapor in the air, xylem

Liquid water in tension Wheeler, Stroock. Nature 455, 208 (2008) Abe Stroock Axisymmertic deformation Creasing Cavitation

10 Osmosis in a liquid membrane solutionsolvent osmotic pressure Chemical potential of water in the solution due to osmosis Number of solute molecules, N Volume of the solution, V Volume per solvent molecule, v solute h Chemical potential of water in the solution due to gravity Jacobus H. Van’t Hoff, Osmotic pressure and chemical equilibrium, Nobel Lecture, 13 December 1901 Chemical potential of water in the solvent Van’t Hoff

11 Osmosis in a gel membrane solution solvent Osmosis balanced by gravity Osmosis balanced by elasticity Crosslink

12 Two ways of doing work Condition of equilibrium Weight Gel = Network + M solvent molecules Solvent Pump Helmholtz free energy of gel work done by the weight work done by the pump Equations of state Chemical potential of solvent Force

13 Gels as transducers A swelling gel is blocked by a hard material. Blocking force. Inhomogeneous deformation. Need to formulate boundary-value problems

14 2 ways of doing work P weight ll l gel pump  M L Reference stateCurrent state Helmholtz free energy per volume Equilibrium condition M=0 Pump, 

15 Inhomogeneous field Deformation gradientConcentration Gibbs (1878) Free-energy function Condition of equilibrium pump Weight Gel Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008) 1.How to prescribe ? 2.How to solve boundary-value problems? 3.What boundary-value problems to solve?

16 Free-energy function Free energy of stretching Flory, Rehner, J. Chem. Phys., 11, 521 (1943) Swelling decreases entropy by straightening polymers. Swelling increases entropy by mixing solvent and polymers. Free energy of mixing Free-energy function Paul Flory

17 + = V dry + V sol = V gel Molecular incompressibility Assumptions: –Individual solvent molecule and polymer are incompressible. –Neglect volumetric change due to physical association –Gel has no voids. (a gel is different from a sponge.) v – volume per solvent molecule Wei Hong

18 Stress-strain relation Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)

19 Anisotropic swelling Free swellingUnidirectional swelling A gel imbibes a different amount of solvent under constraint. Treloar, Trans. Faraday Soc. 46, 783 (1950).

20 Inhomogeneous swelling Concentration is inhomogeneous even in equilibrium. Stress is high near the interface (debond, cavitation). Zhao, Hong, Suo, APL 92, (2008 ) Sternstein, J. Macromolol. Soc. Phys. B6, 243 (1972)

21 Finite element method Hong, Liu, Suo, Int. J. Solids Structures 46, 3282 (2009) ABAQUS UMAT posted online Equilibrium condition Legendre transform A gel in equilibrium is analogous to an elastic solid ABAQUS UMAT

22 Swelling-induced buckling Zishun Liu Liu, Hong, Suo, Swaddiwudhipong, Zhang. Computational Materials Science. In press.

23 Gel and nano-rods Experiment: Sidorenko, Krupenin, Taylor, Fratzl, Aizenberg, Science 315, 487 (2007). Theory: Hong, Zhao, Suo, JAP104, (2008). Joanna Aizenberg

24 Critical humidity can be tuned

25 Swelling-induced bifurcation Zhang, Matsumoto, Peter, Lin, Kamien, Yang, Nano Lett. 8, 1192 (2008). Shu Yang

26 Swelling-induced bifurcation Experiment: Zhang, Matsumoto, Peter, Lin, Kamien, Yang, Nano Lett. 8, 1192 (2008). Simulation: Hong, Liu, Suo, Int. J. Solids Structures 46, 3282 (2009)

27 Time-dependent process Shape change: short-range motion of solvent molecules, fast Volume change: long-range motion of solvent molecules, slow Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)

28 Concurrent deformation and migration Deformation of network Local equilibrium Conservation of solvent Rate process pump Weight Gel Nonequilibrium thermodynamics Maurice Biot JAP 12, 155 (1941) Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)

29 ideal kinetic model Diffusion in true quantities Solvent molecules migrate in a gel by self-diffusion Conversion between true and nominal quantities Hong, Zhao, Zhou, Suo, JMPS 56, 1779 (2008)

30 Finite element method for concurrent deformation and migration Zhang, Zhao, Suo, Jiang, JAP 105, (2009) Hanqing Jiang

31 Alginate Hydrogels AAD Irreversible Ionic crosslinks Covalent crosslinks David Mooney

32 Stress-relaxation test Impermeable Rigid plates Gel disk PBS time strain stress time Zhao, Huebsch, Mooney, Suo, J. Appl. Phys. In press.

33 Elasticity, plasticity, fracture Ionic crosslinks Covalent crosslinks Swollen state45% compressive strain50% compressive strain Swollen state15% compressive strain20% compressive strain 10mm

34 Stress Relaxation Strain: 15% Radius: 6mm Covalent Ionic ~10 2

35 Size effect R

36 Gel with covalent crosslinks relaxes stress by migration of water

Indent a gel 37 F water indenter h gel migration of solvent t h F(∞) F t F(0) Indent into a gel by a certain depth Record force as a function of time Yuhang Hu Hu, Zhao, Vlassak, Suo. Applied Physics Letters 96, (2010) Joost Vlassak

Linear poroelasticity 38 Small deformation Force Balance Mass conservation Kinetics Incompressibility of solvent molecules and polymers Free energy Local equilibrium Constitutive relations Biot, JAP 12, 155 (1941). Originally developed for soil. Now adapted for gels.

PDEs 3 poroelastic constants:

Poisson’s ratio of a gel Initial stateInstantaneous deformation Equilibrium state

covalently crosslinked alginate hydrogel 41 viscoelasticity Poroelasticity Hu, Zhao, Vlassak, Suo. Applied Physics Letters 96, (2010)

42 From relaxation curve to poroelastic constants 2a R F h (a) Plane-strain cylindrical indenter F R h 2a (b) Cylindrical punch 2a θ F h (c) Conical indenter 2a F h R (d) Spherical indenter Hui, Lin, Chuang, Shull, Ling. J. Polymer Sci B: Polymer Phys. 43, 359 (2006) Y.Y. Lin, B.-W. Hu, J. Non-Crystalline Solids (2006). Hu, Zhao, Vlassak, Suo. Applied Physics Letters 96, (2010)

43 Covalently crosslinked alginate gel F(0) F(∞)

44 Crease Liang Fen ( 凉粉 ), a starch gel A food popular in northern China

45 Dough Zhigang, I was making bread this weekend, and realized that when the rising dough was constrained by the bowl it formed the creases that you were talking about in New Orleans. -- An from Michael Thouless

46 Brain

47 Face

48 Crease: theory and experiment Gent, Cho, Rubber Chemistry and Technology 72, 253 (1999) Ghatak, Das, PRL 99, (2007) Experiments: bending rods of rubber and gel Biot, Appl. Sci. Res. A 12, 168 (1963). Theory: linear perturbation analysis Maurice Biot Alan Gent

49 What’s wrong with Biot’s theory? Biot’s theory: infinitesimal strain from the state of homogeneous deformation Hohlfeld, Mahadevan (2008): crease is an instability different from that analyzed by Biot. Crease: Large strain from the state of homogeneous deformation Homogeneous deformation Mahadevan

50 An energetic model of crease Hong, Zhao, Suo, Applied Physics Letters 95, (2009) Neo-Hookean material Plane strain conditions

51 Crease under general loading Critical condition for crease Biot crease Incompressibility Hong, Zhao, Suo, Applied Physics Letters 95, (2009)

52 Crease of a gel during swelling Tanaka et al, Nature 325, 796 (1987) Toyoichi Tanaka

53 Crease of a swelling gel substrate gel 1 Experimental data Southern, Thomas, J. Polym. Sci., Part A, 3, 641 (1965) Tanaka, PRL 68, 2794 (1992) Trujillo, Kim, Hayward, Soft Matter 4, 564 (2008) Theories Ryan Hayward swell Hong, Zhao, Suo, Applied Physics Letters 95, (2009) Further development: Huang, Rui Huang

54 Outlook Gels are used in nature and engineering (tissues of animals and plants, tissue engineering, drug delivery, fluidics…). Deformation and transport are concurrent (The structure of the theory is known. Application of the theory just begins.) The field is wide open (characterization, computation, devices, phenomena).